The position of a particle at time \(t\) is given by the relation \({x}({t})=\left(\frac{{v}_0}{\alpha}\right)\left(1-{e}^{-\alpha {t}}\right)\), where \(v_0\)${\mathrm{}}_{}$ is a constant and \(\alpha >0\). The dimensions of \(v_0\) and \(\alpha\) are respectively:
1. \(\left[M^0L^{1}T^{-1}\right]~\text{and}~\left[T^{-1}\right]\)
2. \(\left[M^0L^{1}T^{0}\right]~\text{and}~\left[T^{-1}\right]\)
3. \(\left[M^0L^{1}T^{-1}\right]~\text{and}~\left[LT^{-1}\right]\)
4. \(\left[M^0L^{1}T^{-1}\right]~\text{and}~\left[T\right]\)

If force (\(F\)), velocity (\(\mathrm{v}\)), and time (\(T\)) are taken as fundamental units, the dimensions of mass will be:

If dimensions of critical velocity \({v_c}\) of a liquid flowing through a tube are expressed as \(\eta^{x}\rho^yr^{z}\), where \(\eta, \rho~\text{and}~r\) are the coefficient of viscosity of the liquid, the density of the liquid, and the radius of the tube respectively, then the values of \({x},\) \({y},\) and \({z},\) respectively, will be:

Planck's constant (\(h\)), speed of light in the vacuum (\(c\)), and Newton's gravitational constant (\(G\)) are the three fundamental constants. Which of the following combinations of these has the dimension of length?

A screw gauge has some zero error but its value is unknown. We have two identical rods. When the first rod is inserted in the screw, the state of the instrument is shown by diagram (I). When both the rods are inserted together in series then the state is shown by the diagram (II). What is the zero error of the instrument? \(1~\text{msd}= 100~\text{csd}=1~\text{mm}\)

Consider a screw gauge without any zero error. What will be the final reading corresponding to the final state as shown?
It is given that the circular head translates \(P\) MSD in \({N}\) rotations. (\(1\) MSD \(=\) \(1~\text{mm}\).)

              
1. \( \left(\frac{{P}}{{N}}\right)\left(2+\frac{45}{100}\right) \text{mm} \)
2. \( \left(\frac{{N}}{{P}}\right)\left(2+\frac{45}{{N}}\right) \text{mm} \)
3. \(P\left(\frac{2}{{N}}+\frac{45}{100}\right) \text{mm} \)
4. \( \left(2+\frac{45}{100} \times \frac{{P}}{{N}}\right) \text{mm}\)

In certain vernier callipers, \(25\) divisions on the vernier scale have the same length as \(24\) divisions on the main scale. One division on the main scale is \(1\) mm long. The least count of the instrument is:

In a vernier calliper, \(N\) divisions of vernier scale coincide with (\(N\text-1\)) divisions of the main scale (in which the length of one division is \(1\) mm). The least count of the instrument should be:
1. \(N~\text{mm}\)
2. \((N-1)~\text{mm}\)
3. \(\frac{1}{10N}~\text{cm}\)
4. \(\frac{1}{(N-1)}~\text{mm}\)

For the expression, \(10^{(at+3)}\), the dimensions of \(a\) will be:
1. \(\left[M^0L^0T^{0}\right]\)
2. \(\left[M^0L^0T^{1}\right]\)
3. \(\left[M^0L^0T^{-1}\right]\)
4. None of these